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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
L[y]=y^{(n)}+a_1 y^{(n-1)}+\cdots+a_{n-1} y^{\prime}+a_n y=0.
\end{equation*}
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<p class="continuation">We seek a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y=e^{r x}.
\end{equation*}
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<p class="continuation">Substituting it into the ODE,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp; r^n e^{r x}+a_1 r^{n-1} e^{r x}+\cdots+a_{n-1} r e^{r x}+a_n e^{r x}=0\\
&amp;\to r^n+a_1 r^{n-1}+\cdots+a_{n-1} r+a_n=0.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">This equation is called the characteristic equation and it has <span class="process-math">\(n\)</span> roots, say <span class="process-math">\(r_1, r_2, \cdots, r_n\text{.}\)</span> Then we have solutions</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
e^{r_1 x},\quad e^{r_2 x},\cdots, e^{r_n x}.
\end{equation*}
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<span class="incontext"><a href="sec4_2.html#p-155" class="internal">in-context</a></span>
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